Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability

Abstract We consider the Lagrangian dynamical system forced to move on a submanifold G_ (qA) =0 G α (q A) = 0. If for some reason we are interested in knowing the dynamics of all original variables qA (t) q A (t), the most economical would be a Hamiltonian formulation on the intermediate phase-space submanifold spanned by reducible variables qA q A and an irreducible set of momenta pᵢ p i, i=A- i = A - α. We describe and compare two different possibilities for establishing the Poisson structure and Hamiltonian dynamics on an intermediate submanifold: Hamiltonian reduction of the Dirac bracket and intermediate formalism. As an example of the application of intermediate formalism, we deduce on this basis the Euler–Poisson equations of a spinning body, establish the underlying Poisson structure, and write their general solution in terms of the exponential of the Hamiltonian vector field.